Square Tricks: Two, Three & Four digit numbers

The square tricks explained in this section will help you find the square of two, three and four digit numbers.

Using a Vedic Math Technique, we will first learn a trick to find the square of any two digit number. We will then apply this trick to find the square of three and four digit numbers.

The Vedic Math Technique that we are going to use here is known as Duplex.

Duplex Terminology

According to duplex terminology,

Duplex of single digit number, say ‘a’ is given by
D(a) = $a^2$

Duplex of double digit number, say ‘ab’ is given by
D(ab) = 2ab

Given below is the table containing the duplex of some numbers

Now with this knowledge of Duplexes, we will see how we can find square of two digit numbers easily.

Trick to find Square of Two Digit Numbers

Using the Vedic math trick explained above lets calculate the square of two digit numbers. At the end of this section you will be able to square any number up to 100 in your head within seconds.

Once you understand this shortcut you can easily extend it to find the squares of three and four digit numbers as explained in the subsequent sections below.

Now, lets get started. You can check out the video below, where we tried to demonstrate this trick in an intuitive way that may find it easier to understand than through a normal learning through reading experience.

Math Trick to find Square of Two Digit Numbers:

Consider a general two digit number, say, ‘ab’.

The square of ‘ab’ will have three parts.

ab² = left most part| middle part | right most part

During calculations, we shall pass from the leftmost duplex to the leftmost duplex.

The left most part will be duplex of ‘a’, the middle part will be duplex of ‘ab’, finally the right most part will be duplex of ‘b’.

i.e ab² = D(a)| D(ab) | D(b)

= a² | 2ab | b²

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[toggle title=”Example 1: Find the square of 12″]

12$^2$ = D(1)| D(12) | D(2)

= 1$^2$ | 2x1x2 | 2$^2$

= 1 | 4 | 4

Hence, 12 = 144 using the duplex methodology

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[toggle title=”Example 2: Find the square of 23″]

23$^2$ = D(2)| D(23) | D(3)

= 2$^2$ | 2x2x3 | 3$^2$

= 4 | 12 | 9

Here, the middle portion has more than two digits, please note that only the leftmost part can have more than one digit. For the rest of the parts, we need to carry over the number preceding the units digit, to the immediate left part , and add it there respectively.

Hence for 12  which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 4 to 5. Therefore,

23$^2$ = 4+1 | 2 | 9

Hence, 23$^2$ = 529 using the duplex methodology

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[toggle title=”Example 2: Find 25 square”]

25$^2$ = D(2)| D(25) | D(5)

= 2$^2$ | 2x2x5 | 5$^2$

= 4 | 20 | 25

Here the excess digit 2 from middle part will be carried over to the left, while the 2 from the right most part will be carried over to the middle part. Therefore,

25$^2$ = 4+2 | 0+2 | 5

Hence, 25$^2$ = 625 using the duplex methodology.

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To practice two digit squares, check out our exercises in Two Digit Squares

Trick to find Square of Three Digit Numbers

Now lets learn the trick to find square of 3 digit number using Vedic Math. At the end of this section you will be able to square numbers from 1 to 1000.

The shortcut to find the square of 3 digit numbers involves using the duplex methodology as explained above. To apply this shortcut, we will also need to find the duplex of 3 digit numbers in addition to the duplex of single and double digit numbers as described above.

Duplex of 3 digit numbers

For a number with three digits, say ‘abc’,

Duplex of ‘abc’ is given by D(abc) = 2ac + $b^2$

For example,

1. Duplex of 125 => D(125) = 2x1x 5 + $2^2$ = 14
2. Duplex of 756 => D(756) = 2x7x6 + $5^2$ = 109

Now with this knowledge of Duplexes, we will see the shortcut on how we can find the square of three digit numbers.

Math Trick to find Square of Three Digit Numbers:

Consider a general three digit number, say, ‘abc’.

The square of ‘abc’ will have five parts as shown below(each part numbered with a digit for our convenience)

abc² = 5 | 4 | 3 | 2 | 1

During calculations, we shall pass from the rightmost duplex to the leftmost duplex.

The rightmost part(1) will be duplex of ‘c’, the next part(2) will be duplex of bc, the middle part(3) will be duplex of ‘abc’, the next part(4) will be duplex of ab and finally the left most part(5) will be duplex of ‘a’.

i.e abc² = D(a) | D(ab) | D(abc)| D(bc) | D(c)

Example 4: Find the square of 321

321$^2$ =D(3) | D(32) | D(321)| D(21) | D(1)

= 3$^2$ | 2x3x2 | 2x3x1 + 2$^2$ | 2x2x1 | 1$^2$

= 9 | 12 | 10 | 4 | 1

As mentioned above only the leftmost part can have more than one digit. For the rest of the parts, we need to carry over the number preceding the units digit, to the immediate left part , and add it there respectively.

Hence for 10 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 12 to 13 and for 13 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 9 to 10.

Hence,

321$^2$ = 10 | 3 | 0 | 4 | 1

= 103041

Example 5: Find the square of 791

791² =D(7) | D(79) | D(791)| D(91) | D(1)

= 7$^2$ | 2x7x9 | 2x7x1 + 9$^2$ | 2x9x1 | 1$^2$

= 49 | 126 | 95 | 18 | 1

= 625681

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To practice two digit squares, check out our exercises in Three Digit Squares

Square of Four Digit Numbers

Proceeding on the same methodology as for 2 and 3 digit squares as mentioned above, to quickly square a 4 digit number, we must know duplex of 4 digit number as well.

For a number with four digits, say ‘abcd’, Duplex of ‘abcd’ => D(abcd) = 2ad + 2bc

The square of ‘abcd’ will have seven parts as shown below

abcd² = D(a) | D(ab) | D(abc) | D(abcd)| D(bcd) | D(cd) | D(d)

= a2 | 2ab | 2ac + b2 | 2ad+2bc | 2bd + c2 | 2cd | d²

Example 6: Find the square of 1221

1221$^2$= D(1) | D(12) | D(122) | D(1221)| D(221) | D(21) | D(1)

= 1$^2$ | 2x1x2 | 2x1x2 + 2$^2$ | 2x1x1+2x2x2 | 2x2x1 + 2$^2$ | 2x2x1 | 1$^2$

= 1 | 4 | 8 | 10 | 8 | 4 | 1

As mentioned above only the leftmost part can have more than one digit. For the rest of the parts, we need to carry over the number preceding the units digit, to the immediate left part , and add it there respectively.

Hence for 10 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 8 to 9

1221$^2$ = 1 | 4 | 9 | 0 | 8 | 4 | 1

= 1490841

Example 7: Find the square of 9654

9654$^2$ = D(9) | D(96) | D(965) | D(9654)| D(654) | D(54) | D(4)

= 9$^2$ | 2x9x6 | 2x9x5 + 6$^2$ | 2x9x4+2x6x5 | 2x6x4 + 5$^2$ | 2x5x4 | 4$^2$

= 81 | 108 | 126 | 132 | 73 | 40 | 16

= 93199716

Do try our android app – Math Tricks Workout. The app is developed to improve mental arithmetic using a series of left to right fast math workouts.

Scan the QR code below or click on it for more details.